The Oscillation Function on Metric Spaces

Duszyński, Zbigniew

Real Anal. Exchange, Tome 25 (1999) no. 1, p. 489-492 / Harvested from Project Euclid

Résumé

For each metric space $(X,\xi)$ and each bounded function $f\colon X\to \R$ the family of the sets $\O_f(y)=\{x\in X\colon\, \om_f(x)\ge y\}$ ($\om_f(x)$ is the oscillation of $f$) has some well known properties. In this paper it is constructively shown that for each family $\{\O(y)\}_{y\in [0,1]}$ of subsets of $X$ (separable and $\C$-dense in itself) having similar properties there exists a function $f\colon\, X\to [0,1]$ such that $\O_f(y)=\O(y)$ for each $y\in [0,1]$.

Détail

Publié le : 1999-05-15
Classification:  oscillation function,  separable metric space,  26A15